# Category:Definitions/Borel Sigma-Algebras

This category contains definitions related to Borel Sigma-Algebras.

Related results can be found in Category:Borel Sigma-Algebras.

### Topological Space

Let $\struct {S, \tau}$ be a topological space

The **Borel sigma-algebra** $\map \BB {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$.

That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.

### Metric Space

Let $\struct {S, \norm {\, \cdot \,} }$ be a metric space.

The **Borel sigma-algebra** (or **$\sigma$-algebra**) on $\struct {S, \norm {\, \cdot \,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset S$.

By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on a topological space.

## Pages in category "Definitions/Borel Sigma-Algebras"

The following 5 pages are in this category, out of 5 total.